List of Fourier analysis topics
Encyclopedia
This is an alphabetical list of Fourier analysis topics. See also the list of Fourier-related transforms, and the list of harmonic analysis topics.
- Almost periodic functionAlmost periodic functionIn mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov,...
- ATS theoremATS theoremIn mathematics, the ATS theorem is the theorem on the approximation of atrigonometric sum by a shorter one. The application of the ATS theorem in certain problems of mathematical and theoretical physics can be very helpful....
- AutocorrelationAutocorrelationAutocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them...
- AutocovarianceAutocovarianceIn statistics, given a real stochastic process X, the autocovariance is the covariance of the variable with itself, i.e. the variance of the variable against a time-shifted version of itself...
- Banach algebraBanach algebraIn mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
- Bessel functionBessel functionIn mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
- Compact groupCompact groupIn mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...
- Continuous Fourier transformContinuous Fourier transformThe Fourier transform is a mathematical operation that decomposes a function into its constituent frequencies, known as a frequency spectrum. For instance, the transform of a musical chord made up of pure notes is a mathematical representation of the amplitudes of the individual notes that make...
- Convergence of Fourier seriesConvergence of Fourier seriesIn mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics...
- ConvolutionConvolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
- Convolution theoremConvolution theoremIn mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain equals point-wise multiplication in the other domain...
- DFT matrixDFT matrixA DFT matrix is an expression of a discrete Fourier transform as a matrix multiplication.-Definition:An N-point DFT is expressed as an N-by-N matrix multiplication as X = W x, where x is the original input signal, and X is the DFT of the signal.The transformation W of size N\times N can be defined...
- Dini testDini testIn mathematics, the Dini and Dini-Lipschitz tests are highly precise tests that can be used to prove that the Fourier series of a function converges at a given point. These tests are named after Ulisse Dini and Rudolf Lipschitz.- Definition :...
- Dirichlet kernel
- Dirichlet problemDirichlet problemIn mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of the region....
- Discrete Fourier transformDiscrete Fourier transformIn mathematics, the discrete Fourier transform is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function...
, Discrete Fourier seriesDiscrete Fourier seriesA Fourier series is a representation of a function in terms of a summation of an infinite number of harmonically-related sinusoids with different amplitudes and phases. The amplitude and phase of a sinusoid can be combined into a single complex number, called a Fourier coefficient. The Fourier... - Discrete Hartley transformDiscrete Hartley transformA discrete Hartley transform is a Fourier-related transform of discrete, periodic data similar to the discrete Fourier transform , with analogous applications in signal processing and related fields. Its main distinction from the DFT is that it transforms real inputs to real outputs, with no...
- DistributionDistribution (mathematics)In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
- Fast cosine transform
- Fast Fourier transformFast Fourier transformA fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...
- Fejér kernel
- Fourier amplitude sensitivity testingFourier amplitude sensitivity testingFourier amplitude sensitivity testing is a variance-based global sensitivity analysis method. The sensitivity value is defined based on conditional variances which indicate the individual or joint effects of the uncertain inputs on the output....
- Fourier integral operatorFourier integral operatorIn mathematical analysis, Fourier integral operators have become an important tool in the theory of partial differential equations. The class of Fourier integral operators contains differential operators as well as classical integral operators as special cases....
- Fourier inversion theoremFourier inversion theoremIn mathematics, Fourier inversion recovers a function from its Fourier transform. Several different Fourier inversion theorems exist.Sometimes the following expression is used as the definition of the Fourier transform:...
- Fourier operatorFourier operatorThe Fourier operator is the kernel of the Fredholm integral of the first kind that defines the continuous Fourier transform.It may be thought of as a limiting case for when the size of the discrete Fourier transform increases without bound while its spatial resolution also increases without bound,...
- Fourier opticsFourier opticsFourier optics is the study of classical optics using Fourier transforms and can be seen as the dual of the Huygens-Fresnel principle. In the latter case, the wave is regarded as a superposition of expanding spherical waves which radiate outward from actual current sources via a Green's function...
- Fourier seriesFourier seriesIn mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...
- Fourier shell correlation
- Fourier transformFourier transformIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
- Fourier transform on finite groups
- Fractional Fourier transformFractional Fourier transformIn mathematics, in the area of harmonic analysis, the fractional Fourier transform is a linear transformation generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power where n need not be an integer — thus, it can transform a function to an...
- Frequency spectrumFrequency spectrumThe frequency spectrum of a time-domain signal is a representation of that signal in the frequency domain. The frequency spectrum can be generated via a Fourier transform of the signal, and the resulting values are usually presented as amplitude and phase, both plotted versus frequency.Any signal...
- Gabor atom
- Generalized Fourier seriesGeneralized Fourier seriesIn mathematical analysis, many generalizations of Fourier series have proved to be useful.They are all special cases of decompositions over an orthonormal basis of an inner product space....
- Gibbs phenomenonGibbs phenomenonIn mathematics, the Gibbs phenomenon, named after the American physicist J. Willard Gibbs, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity: the nth partial sum of the Fourier series has large...
- Haar measureHaar measureIn mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
- Hardy spaceHardy spaceIn complex analysis, the Hardy spaces Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper...
- Harmonic analysisHarmonic analysisHarmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
- Harmonic functionHarmonic functionIn mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
- Laplace equation
- Laplace operatorLaplace operatorIn mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...
- Laplace transform
- List of cycles, a very interdisciplinary article
- List of Fourier-related transforms
- LTI system theoryLTI system theoryLinear time-invariant system theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and time-invariant...
- Marcinkiewicz theoremMarcinkiewicz theoremIn mathematics, the Marcinkiewicz interpolation theorem, discovered by , is a result bounding the norms of non-linear operators acting on Lp spaces....
- Modulus of continuityModulus of continuityIn mathematical analysis, a modulus of continuity is a function\omega:[0,\infty]\to[0,\infty]used to measure quantitatively the uniform continuity of functions. So, a function f:I\to\R admits \omega as a modulus of continuity if and only if|f-f|\leq\omega,for all x and y in the domain of f...
- Multiplier (Fourier analysis)Multiplier (Fourier analysis)In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions. These operators act on a function by altering its Fourier transform. Specifically they multiply the Fourier transform of a function by a specified function known as the multiplier or symbol...
- Non-uniform discrete Fourier transformNon-uniform discrete Fourier transformIn applied mathematics, the non-uniform discrete Fourier transform of a signal is a type of Fourier transform, related to a discrete Fourier transform or discrete-time Fourier transform, but in which the input signal is not sampled at equally-spaced intervals. As a result of this, the computed...
- Nyquist–Shannon sampling theoremNyquist–Shannon sampling theoremThe Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...
- oscillatory integralOscillatory integralIn mathematical analysis an oscillatory integral is a type of distribution. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals...
- oscillatory integral operator
- Paley–Wiener theoremPaley–Wiener theoremIn mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley and Norbert Wiener . The original theorems did not use the language of distributions,...
- Parseval's theoremParseval's theoremIn mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later...
- Periodic functionPeriodic functionIn mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
- Peter–Weyl theoremPeter–Weyl theoremIn mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group G...
- Pinsky phenomenonPinsky phenomenonThe Pinsky phenomenon is a result in Fourier analysis, a branch of mathematics . This phenomenon was discovered by Mark Pinsky of Northwestern University in Evanston, Illinois...
- Plancherel theoremPlancherel theoremIn mathematics, the Plancherel theorem is a result in harmonic analysis, proved by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum....
- Poisson summation formulaPoisson summation formulaIn mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples...
- Pontryagin dualityPontryagin dualityIn mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...
- Projection-slice theoremProjection-slice theoremIn mathematics, the projection-slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal:...
- Regressive discrete Fourier seriesRegressive discrete fourier seriesIn applied mathematics, the regressive discrete Fourier series is a generalization of the discrete Fourier transform where the Fourier series coefficients are computed in a least squares sense and the period is arbitrary, i.e., not necessarily equal to the length of the data. It was first...
- Riesz–Thorin theorem
- Set of uniquenessSet of uniquenessIn mathematics, a set of uniqueness is a concept relevant to trigonometric expansions which are not necessarily Fourier series. Their study is a relatively pure branch of harmonic analysis.- Definition :...
- Sigma approximationSigma approximationIn mathematics, σ-approximation adjusts a Fourier summation to eliminate the Gibbs phenomenon which would otherwise occur at discontinuities.A σ-approximated summation for a series of period T can be written as follows:...
- Sine and cosine transformsSine and cosine transformsIn mathematics, the Fourier sine and cosine transforms are special cases of thecontinuous Fourier transform, arising naturally when attempting to transform odd and even functions, respectively.The general Fourier transform is defined as:...
- Sobolev spaceSobolev spaceIn mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space...
- Spectrum continuation analysisSpectrum continuation analysisSpectrum continuation analysis is a generalization of the concept of Fourier series to non-periodic functions of which only a fragment has been sampled in the time domain....
- Spherical harmonicSpherical HarmonicSpherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. It tells the story of Dyhianna Selei , the Ruby Pharaoh of the Skolian Imperialate, as she strives to reform her government and reunite her family in the aftermath of a devastating interstellar...
- Standing wave discrete Fourier transform
- Topological groupTopological groupIn mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
- Uncertainty principle for the short-time Fourier transformUncertainty principle for the short-time Fourier transformThere are many things one can do to signals to study them. However, if one do something to a signal that modifies it in some way, one should not confuse uncertainty principle applied to the modified signal with the uncertainty principle as applied to the original signal...
- Unit circleUnit circleIn mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
- Unit disc
- Whittaker–Shannon interpolation formulaWhittaker–Shannon interpolation formulaThe Whittaker–Shannon interpolation formula or sinc interpolation is a method to reconstruct a continuous-time bandlimited signal from a set of equally spaced samples.-Definition:...